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Table T(n,k) = n*(n+2^k-1)/2, n, k > 0 read by antidiagonals.
1

%I #26 Jan 05 2018 14:45:22

%S 1,2,3,4,5,6,8,9,9,10,16,17,15,14,15,32,33,27,22,20,21,64,65,51,38,30,

%T 27,28,128,129,99,70,50,39,35,36,256,257,195,134,90,63,49,44,45,512,

%U 513,387,262,170,111,77,60,54,55,1024,1025,771,518,330,207,133,92,72,65,66

%N Table T(n,k) = n*(n+2^k-1)/2, n, k > 0 read by antidiagonals.

%C Column number 1 A000217 n*(n+1)/2,

%C column number 2 A000096 n*(n+3)/2,

%C column number 3 A055999 n*(n+7)/2,

%C column number 4 A056121 n*(n+15)/2,

%C column number 5 A132758 n*(n+31)/2.

%C Row number 1 A000079 2^k,

%C row number 2 A000051 2^k + 1.

%H Boris Putievskiy, <a href="/A209274/b209274.txt">Rows n = 1..140 of triangle, flattened</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%F a(n) = A002260(n)*(A002260(n)+2^A004736(n)-1)/2.

%F a(n) = i*(i+2^j-1)/2,

%F where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

%e The start of the sequence as table:

%e 1....2...4...8...16...32...64...

%e 3....5...9..17...33...65..129...

%e 6....9..15..27...51...99..195...

%e 10..14..22..38...70..134..262...

%e 15..20..30..50...90..170..330...

%e 21..27..39..63..111..207..399...

%e 28..35..49..77..133..245..469...

%e . . .

%e The start of the sequence as triangle array read by rows:

%e 1;

%e 2,3;

%e 4,5,6;

%e 8,9,9,10;

%e 16,17,15,14,15;

%e 32,33,27,22,20,21;

%e 64,65,51,38,30,27,28;

%e . . .

%e Row number r contains r numbers.

%t b[n_] := n - d[n]*(d[n] + 1)/2; c[n_] := (d[n]^2 + 3*d[n] + 4)/2 - n; d[n_] := Floor[(-1 + Sqrt[8*n - 7])/2]; a[n_] := b[n]*(b[n] + 2^c[n] - 1)/2; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Jan 04 2018 *)

%o (Python)

%o t=int((math.sqrt(8*n-7) - 1)/ 2)

%o i=n-t*(t+1)/2

%o j=(t*t+3*t+4)/2-n

%o result = i*(i+2**j-1)/2

%o (PARI) a(n, k) = n*(n+2^k-1)/2

%o array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))

%o /* Print initial 7 rows and 8 columns of table as follows */

%o array(7, 8) \\ _Felix Fröhlich_, Jan 05 2018

%Y Cf. A000217, A000096, A055999, A056121, A132758, A000079, A000051.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Jan 15 2013